Motivic Chern classes of Schubert cells, Hecke algebras, and applications to Casselman's problem

Abstract

Motivic Chern classes are elements in the K-theory of an algebraic variety X, depending on an extra parameter y. They are determined by functoriality and a normalization property for smooth X. In this paper we calculate the motivic Chern classes of Schubert cells in the (equivariant) K-theory of flag manifolds G/B. We show that the motivic class of a Schubert cell is determined recursively by the Demazure-Lusztig operators in the Hecke algebra of the Weyl group of G, starting from the class of a point. The resulting classes are conjectured to satisfy a positivity property. We use the recursions to give a new proof that they are equivalent to certain K-theoretic stable envelopes recently defined by Okounkov and collaborators, thus recovering results of Fehér, Rimányi and Weber. The Hecke algebra action on the K-theory of the Langlands dual flag manifold matches the Hecke action on the Iwahori invariants of the principal series representation associated to an unramified character for a group over a nonarchimedean local field. This gives a correspondence identifying the Poincaré duals of the motivic Chern classes to the standard basis in the Iwahori invariants, and the fixed point basis to Casselman's basis. We apply this correspondence to prove two conjectures of Bump, Nakasuji and Naruse concerning factorizations and holomorphy properties of the coefficients in the transition matrix between the standard and the Casselman's basis. 1 The class SM Cy(Ω) := D(M Cy (Ω)) λy(T * (G/B)) may be regarded as the motivic analogue of the Segre-MacPherson class c * (1 1 Ω ) c(T (G/B)) .

Characteristic classes of Borel orbits of square-zero upper-triangular matrices

“…The operator A i has appeared in [AMSS17, §5] (denoted by L ∨ ). The K-theoretic version is present in [AMSS19] (denoted by T ∨ ). In both cases these operators are dual to those computing the classes of the open orbits.…”

“…Here e(N ) and e K (N ) stand for the equivariant Euler classes in cohomology and K-theory. The operators appearing in the theorem are versions of Hecke operators from [AMSS17] or [AMSS19].…”

Wall-crossings and a categorification of $K$-theory stable bases of the Springer resolution

“…With this, in [SZZ17] we provided a K-theoretic interpretation of the Macdonald's formula for the spherical function [C80] and the Casselman-Shalika formula for the spherical Whittaker function [CS80]. Pulling back the stable basis to the flag variety from its cotangent bundle, we get the motivic Chern classes [BSY10] of the Schubert cells [AMSS19,FRW18]. This connection is used to prove a series of conjectures about the Casselman basis, see [AMSS19,BN11,BN17].…”

“…Pulling back the stable basis to the flag variety from its cotangent bundle, we get the motivic Chern classes [BSY10] of the Schubert cells [AMSS19,FRW18]. This connection is used to prove a series of conjectures about the Casselman basis, see [AMSS19,BN11,BN17].…”

Motivic Chern classes of configuration spaces

“…the Chern-Schwartz-MacPherson class [Mac74], [Ohm06] for the equivariant version). Lately its equivariant counterpart was defined in [FRW18b,AMSS19] and studied for Schubert varieties in flag varieties. Consider an algebraic torus T. The motivic Chern class mC T assigns to every T equivariant map f : X → M of T-varieties an element of G T (M)[y], i.e.…”